<head>
<title>Lambert Conic Conformal (2SP)</title>
</head>
<body>

<h1>Lambert Conic Conformal (2SP)</h1>

<table border>

<td>Name
<td>Lambert Conic Conformal (2SP)
<tr>

<td>EPSG Code
<td>9802
<tr>

<td>GeoTIFF Code
<td>CT_LambertConfConic_2SP (8)
<tr>

<td>
<td>CT_LambertConfConic (8)
<tr>

<td>OGC WKT Name
<td> Lambert_Conformal_Conic_2SP
<tr>

<td>Supported By
<td>EPSG, GeoTIFF, PROJ.4, OGC WKT
<tr>

</table>

<h3>Projection Parameters</h3>

<table border>
<th>Name
<th>EPSG #
<th>GeoTIFF ID
<th>OGC WKT
<th>Units
<th>Notes

<tr>
<td>Latitude of false origin
<td> 1
<td> FalseOriginLat
<td> latitude_of_origin
<td> Angular
<td>

<tr>
<td> Longitude of false origin
<td> 2
<td> FalseOriginLong
<td> central_meridian
<td> Angular
<td>

<tr>
<td>Latitude of first standard parallel
<td> 3
<td> StdParallel1
<td> standard_parallel_1
<td>Angular
<td>

<tr>
<td>Latitude of second standard parallel
<td> 4
<td> StdParallel2
<td> standard_parallel_2
<td>Angular
<td>

<tr>
<td>Easting of false origin
<td>6
<td>FalseOriginEasting
<td>false_easting
<td>Linear
<td>

<tr>
<td>Northing of false origin
<td>7
<td>FalseOriginNorthing
<td>false_northing
<td>Linear
<td>

</table>

<h3>PROJ.4 Organization</h3>

<b>
<pre>
  +proj=lcc   +lat_1=<i>Latitude of first standard parallel</i>
              +lat_2=<i>Latitude of second standard parallel</i>
              +lat_0=<i>Latitude of false origin</i> 
              +lon_0=<i>Longitude of false origin</i>
              +x_0=<i>False Origin Easting</i>
              +y_0=<i>False Origin Northing</i>
</pre>
</b>

<h3>EPSG Notes</h3>

Conical projections with one standard parallel are normally considered to maintain the 
nominal map scale along the parallel of latitude which is the line of contact between the 
imagined cone and the ellipsoid. For a one standard parallel Lambert the natural origin of 
the projected coordinate system is the intersection of the standard parallel with the 
longitude of origin (central meridian). See Figure 5. To maintain the conformal property 
the spacing of the parallels is variable and increases with increasing distance from the 
standard parallel, while the meridians are all straight lines radiating from a point on the 
prolongation of the ellipsoid's minor axis. <p>

Sometimes however, although a one standard parallel Lambert is normally 
considered to have unity scale factor on the standard parallel, a scale factor of 
slightly less than unity is introduced on this parallel. This is a regular feature of the 
mapping of some former French territories and has the effect of making the scale 
factor unity on two other parallels either side of the standard parallel. The projection 
thus, strictly speaking, becomes a Lambert Conic Conformal projection with two 
standard parallels. From the one standard parallel and its scale factor it is possible to 
derive the equivalent two standard parallels and then treat the projection as a two 
standard parallel Lambert conical conformal, but this procedure is seldom adopted. 
Since the two parallels obtained in this way will generally not have integer values of 
degrees or degrees and minutes it is instead usually preferred to select two specific 
parallels on which the scale factor is to be unity, - as for several State Plane 
Coordinate systems in the United States.<p>
	
 The choice of the two standard parallels will usually be made according to the latitudinal 
extent of the area which it is wished to map, the parallels usually being chosen so that they 
each lie a proportion inboard of the north and south margins of the mapped area. Various 
schemes and formulas have been developed to make this selection such that the maximum 
scale distortion within the mapped area is minimised, e.g. Kavraisky in 1934, but whatever 
two standard parallels are adopted the formulas for the projected coordinates are the same. <p>

For territories with limited latitudinal extent but wide longitudinal width it may 
sometimes be preferred to use a single projection  rather than several bands or zones 
of a Transverse Mercator. If the latitudinal extent is also large there may still be a 
need to use two or more zones if the scale distortion at the extremities of the one 
zone becomes too large to be tolerable.<p>

To derive the projected Easting and Northing coordinates of a point with 
geographical coordinates (*,*) the formulas for the two standard parallel case are:<p>

<pre>
	Easting, E = EF + r sin *
	Northing, N = NF + rF - r cos *	 

where 	m = cos*/(1 - e2sin2*)1/2     for m1, *1, and m2, *2 where *1 and *2  are the latitudes 
of 			the standard parallels
	t  = tan(*/4 - */2)/[(1 - e sin*)/(1 + e sin*)]e/2   for t1, t2, tF and t using *1,*2,*
F and * 			respectively
	n = (loge m1 - loge m2)/(loge t1 - loge t2)
	F = m1/(nt1n)
	r =  a F tn         for rF and r, where rF is the radius of the parallel of latitude of the 
false origin
	* = n(* - *0)

The reverse formulas to derive the latitude and longitude of a point from its Easting and 
Northing values are:

	* = */2 - 2arctan{t'[(1 - esin*)/(1 + esin*)]e/2}
	* = *'/n +*0
where
	r' = *[(E - EF)2 + {rF - (N - NF)}2]1/2 , taking the sign of n
	t' = (r'/aF)1/n
	*' = arctan [(E- EF)/(rF - (N- NF))]
and n, F, and rF are derived as for the forward calculation.

With minor modifications these formulas can be used for the single standard parallel 
case. Then
	E = FE + r sin*
	N = FN + r0 - r cos*,  using the natural origin rather than the false origin.
where
	n = sin *0
	r = aFtn k0     	for r0, and r
	t is found for  t0, *0 and t, * and m, F, and * are found as for the two standard 
parallel case
	The reverse formulas for * and * are as for the two standard parallel case above, 
with n, F and r0 as before and

	*' = arctan[(E - FE)/{r0 -(N - FN)}]
	r' = *[(E - FE)2 + {r0 - (N - FN)}2]1/2
	t' = (r'/ak0F)1/n","For Projected Coordinate System NAD27 / Texas South Central

Parameters:
Ellipsoid  Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet
                                   1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866

First Standard Parallel          28o23'00""N  =   0.49538262 rad
Second Standard Parallel    30o17'00""N  =   0.52854388 rad
Latitude False Origin            27o50'00""N  =   0.48578331 rad
Longitude False Origin         99o00'00""W = -1.72787596 rad
Easting at false origin           2000000.00  US survey feet
Northing at false origin          0.00  US survey feet

Forward calculation for: 
Latitude       28o30'00.00""N  =  0.49741884 rad
Longitude    96o00'00.00""W = -1.67551608 rad

first gives :
m1    = 0.88046050      m2 = 0.86428642
t        = 0.59686306      tF  = 0.60475101
t1      = 0.59823957      t2 = 0.57602212
n       = 0.48991263       F = 2.31154807
r        = 37565039.86    rF = 37807441.20
theta = 0.02565177

Then Easting E =      2963503.91 US survey feet
         Northing N =      254759.80 US survey feet

Reverse calculation for same easting and northing first gives:
theta' = 0.025651765     r' = 37565039.86
t'        = 0.59686306

Then Latitude     	= 28o30'00.000""N
         Longitude   = 96o00'00.000""W
</pre>

</body>
